GATE 2023 BT – Question 57
A protein has three identical sites at the vertices of an equilateral triangle. One donor has quantum yield $\phi_D=0.5$. Adding one acceptor reduces it to 0.25. With one donor and two acceptors, the donor quantum yield (three decimal places) is ___________.
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Correct answer: 0.157 to 0.177
Explanation
**FRET and donor quantum yield.** The donor can lose its excitation by emission (rate $k_r$), by non-radiative decay ($k_{nr}$), and by energy transfer to an acceptor ($k_T$ for each acceptor). The quantum yield is
$$\phi=\frac{k_r}{k_r+k_{nr}+n\,k_T},$$
where $n$ is the number of acceptors.
**Step 1: no acceptor.** $\phi_D=\dfrac{k_r}{k_r+k_{nr}}=0.5$. Call the total decay rate without acceptors $k_0=k_r+k_{nr}$.
**Step 2: one acceptor.** $\phi=\dfrac{k_r}{k_0+k_T}=0.25$. Since $k_r/k_0=0.5$, the denominator must be doubled: $k_0+k_T=2k_0$, so $k_T=k_0$.
**Step 3: two acceptors.** The sites sit at the corners of an equilateral triangle, so both acceptors are at the same distance from the donor and each has the same $k_T$:
$$\phi=\frac{k_r}{k_0+2k_T}=\frac{k_r}{3k_0}=\frac{0.5}{3}=0.1667\approx\mathbf{0.167}.$$